scholarly journals MAXIMUM ENTROPY FOR REDUCED MOMENT PROBLEMS

2000 ◽  
Vol 10 (07) ◽  
pp. 1001-1025 ◽  
Author(s):  
MICHAEL JUNK
Keyword(s):  
Maximum Entropy ◽  
Wide Class ◽  
Moment Problem ◽  
Entropy Solution ◽  
Unbounded Domains ◽  
Entropy Solutions ◽  
Moment Problems ◽  
Precise Condition ◽  
The Moment

The existence of maximum entropy solutions for a wide class of reduced moment problems on arbitrary open subsets of ℝd is considered. In particular, new results for the case of unbounded domains are obtained. A precise condition is presented under which solvability of the moment problem implies existence of a maximum entropy solution.

Solutions of the Al-Salam–Chihara and allied moment problems

2019 ◽  
Vol 18 (02) ◽  
pp. 185-210 ◽  
Author(s):  
Mourad E. H. Ismail
Keyword(s):  
Moment Problem ◽  
Sufficient Conditions ◽  
Infinite Family ◽  
Weight Functions ◽  
Moment Problems ◽  
Order Operator ◽  
Absolutely Continuous ◽  
Rodrigues Formula ◽  
Wilson Operator ◽  
The Moment

We study the moment problem associated with the Al-Salam–Chihara polynomials in some detail providing raising (creation) and lowering (annihilation) operators, Rodrigues formula, and a second-order operator equation involving the Askey–Wilson operator. A new infinite family of weight functions is also given. Sufficient conditions for functions to be weight functions for the [Formula: see text]-Hermite, [Formula: see text]-Laguerre and Stieltjes–Wigert polynomials are established and used to give new infinite families of absolutely continuous orthogonality measures for each of these polynomials.

scholarly journals New Results on Markov Moment Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Octav Olteanu
Keyword(s):  
Moment Problem ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Moment Problems ◽  
Finite Dimensional ◽  
Integrable Functions ◽  
Truncated Moment Problem ◽  
Markov Moment ◽  
The Moment ◽  
Necessary And Sufficient

The present work deals with the existence of the solutions of some Markov moment problems. Necessary conditions, as well as necessary and sufficient conditions, are discussed. One recalls the background containing applications of extension results of linear operators with two constraints to the moment problem and approximation by polynomials on unbounded closed finite-dimensional subsets. Two domain spaces are considered: spaces of absolute integrable functions and spaces of analytic functions. Operator valued moment problems are solved in the latter case. In this paper, there is a section that contains new results, making the connection to some other topics: bang-bang principle, truncated moment problem, weak compactness, and convergence. Finally, a general independent statement with respect to polynomials is discussed.

On Polynomial Maximum Entropy Method for Classical Moment Problem

2015 ◽  
Vol 8 (1) ◽  
pp. 117-127
Author(s):  
Jiu Ding ◽  
Noah H. Rhee ◽  
Chenhua Zhang
Keyword(s):  
Maximum Entropy ◽  
Moment Problem ◽  
Maximum Entropy Method ◽  
Entropy Method ◽  
Moment Problems ◽  
Monomial Basis ◽  
Ill Conditioning ◽  
Legendre Moment ◽  
Hausdorff Moment Problem ◽  
Classical Moment Problem

AbstractThe maximum entropy method for the Hausdorff moment problem suffers from ill conditioning as it uses monomial basis {1,x,x2,...,xn}. The maximum entropy method for the Chebyshev moment probelm was studied to overcome this drawback in. In this paper we review and modify the maximum entropy method for the Hausdorff and Chebyshev moment problems studied in and present the maximum entropy method for the Legendre moment problem. We also give the algorithms of converting the Hausdorff moments into the Chebyshev and Lengendre moments, respectively, and utilizing the corresponding maximum entropy method.

scholarly journals On the Truncated Multidimensional Moment Problems in Cn

Axioms ◽  
2022 ◽  
Vol 11 (1) ◽  
pp. 20
Author(s):  
Sergey Zagorodnyuk
Keyword(s):  
Integral Representation ◽  
Finite Subset ◽  
Moment Problem ◽  
Complex Numbers ◽  
Moment Problems ◽  
Linear Functionals ◽  
Truncated Moment Problem ◽  
The Moment ◽  
Multidimensional Moment Problem ◽  
Multidimensional Moment Problems

We consider the problem of finding a (non-negative) measure μ on B(Cn) such that ∫Cnzkdμ(z)=sk, ∀k∈K. Here, K is an arbitrary finite subset of Z+n, which contains (0,…,0), and sk are prescribed complex numbers (we use the usual notations for multi-indices). There are two possible interpretations of this problem. Firstly, one may consider this problem as an extension of the truncated multidimensional moment problem on Rn, where the support of the measure μ is allowed to lie in Cn. Secondly, the moment problem is a particular case of the truncated moment problem in Cn, with special truncations. We give simple conditions for the solvability of the above moment problem. As a corollary, we have an integral representation with a non-negative measure for linear functionals on some linear subspaces of polynomials.

scholarly journals Sample Dependence in the Maximum Entropy Solution to the Generalized Moment Problem

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Henryk Gzyl
Keyword(s):  
Probability Density ◽  
Maximum Entropy ◽  
Moment Problem ◽  
Entropy Solution ◽  
Random Variable ◽  
Actual Practice ◽  
Expected Values ◽  
Generalized Moment Problem ◽  
Generalized Moment

The method of maximum entropy is quite a powerful tool to solve the generalized moment problem, which consists in determining the probability density of a random variableXfrom the knowledge of the expected values of a few functions of the variable. In actual practice, such expected values are determined from empirical samples, leaving open the question of the dependence of the solution upon the sample. It is the purpose of this note to take a few steps towards the analysis of such dependence.

ON THE SOLVABILITY OF MAXIMUM ENTROPY MOMENT PROBLEMS IN TEXTURE ANALYSIS

2012 ◽  
Vol 22 (12) ◽  
pp. 1250043 ◽  
Author(s):  
MICHAEL JUNK ◽  
JOHANNES BUDDAY ◽  
THOMAS BÖHLKE
Keyword(s):  
Distribution Function ◽  
Maximum Entropy ◽  
Orientation Distribution ◽  
Moment Problems ◽  
Topological Groups ◽  
Crystallite Orientation ◽  
Locally Compact ◽  
Texture Coefficients ◽  
Moment Functions ◽  
The Moment

The estimation of the crystallite orientation distribution function based on the leading texture coefficients can be rephrased as a maximum entropy moment problem. In this paper, we prove the solvability of these moment problems under quite general assumptions on the moment functions which carries over to general locally compact and σ-compact Hausdorff topological groups.

scholarly journals The Stieltjes condition and multidimensional $${\mathcal {K}}$$-moment problems

2021 ◽  
Author(s):  
Konrad Schmüdgen
Keyword(s):  
Moment Problem ◽  
Moment Problems ◽  
Solvability Theorem ◽  
The Moment

AbstractWe prove a solvability theorem for the Stieltjes problem on $$\mathbb {R}^d$$ R d which is based on the multivariate Stieltjes condition $$\sum _{n=1}^\infty L(x_j^{n})^{-1/(2n)} =+\infty $$ ∑ n = 1 ∞ L ( x j n ) - 1 / ( 2 n ) = + ∞ , $$j=1,\dots ,d.$$ j = 1 , ⋯ , d . This result is applied to derive a new solvability theorem for the moment problem on unbounded semi-algebraic subsets of $$\mathbb {R}^d$$ R d .

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